Continuity mistake: The crew on the destroyer imposing the blockade initially saw the two Russian freighters on their radar screen, and then they saw a third 'blip' on the radar screen as the ships approached the blockade. The radarman identified the first two blips as surface ships, and the third blip as a submarine. Unless that was a sonar display, they would never have seen the submarine. The third contact would have been made by the ship's sonarman, and the two surface contacts would have been identified by the radarman.
Continuity mistake: The Pierce fires its guns to clear the barrels of the already loaded starshell. This is technically correct but when the starshells burst there are four of them i.e. the same as the number of gun barrels on the Pierce. Earlier however, when the Pierce first confronts the Grozny, there is a shot in the 5in gun turret showing at least one barrel being loaded with practice shell (i.e. solid shot). The Captain reports to the Admiral that the guns are loaded.
Continuity mistake: When the President and advisors are sitting around a large table and communicating with the captain of the ship at the beginning of the blockade, when viewed from the front, the President clearly has his right hand up to his chin while concerned and deep in thought. As the camera angle switches to his back several times, his left hand is on his chin. Front angle right, rear angle left.